The symplectic groupoid associated to this is (by the discussion there) supposed to be the fundamental groupoidΠ1(X)\Pi_1(X) of XX equipped on its space of morphisms with the differential form p1*ω−p2*ωp_1^* \omega - p_2^* \omega, where p1,p2p_1,p_2 are the two endpoint projections from paths in XX to XX.

For simplicity we shall start with the simple situation where (X,ω)(X,\omega) has a global Darboux coordinate chart{xi}\{x^i\}. Write {ωij}\{\omega_{i j}\} for the components of the symplectic form in these coordinates, and {ωij}\{\omega^{i j}\} for the components of the inverse.

Then the Chevalley-Eilenberg algebraCE(𝔓)CE(\mathfrak{P}) is generated from {xi}\{x^i\} in degree 0 and {∂i}\{\partial_i\} in degree 1, with differential given by

Over a test space UU in degree 1 an element in exp(𝔓)diff\exp(\mathfrak{P})_{diff} is a pair (Xi,ηi)(X^i, \eta^i)

Xi∈C∞(U×Δ1)
X^i \in C^\infty(U \times \Delta^1)

ηi∈Ωvert1(U×Δ1)
\eta^i \in \Omega^1_{vert}(U \times \Delta^1)

subject to the verticality constraint, which says that along Δ1\Delta^1 we have

dΔ1Xi+ηΔ1i=0.
d_{\Delta^1} X^i + \eta^i_{\Delta^1} = 0
\,.

The vertical morphism exp(𝔓)diff→exp(𝔓)\exp(\mathfrak{P})_{diff} \to \exp(\mathfrak{P}) has in fact a section whose image is given by those pairs for which ηi\eta^i has no leg along UU. We therefore find the desired form on exp(𝔓)\exp(\mathfrak{P}) by evaluating the top morphism on pairs of this form.

However, a Poisson structure on a manifold XX is equivalent to the structure of a Poisson Lie algebroid𝔓\mathfrak{P} over XX. This is noteworthy, because the latter is again symplectic, as a Lie algebroid, even if the underlying Poisson manifold is not symplectic: it is a symplectic Lie algebroid .

Formal hints for such a generalization had been noted in (Ševera), in particular in its concluding table. More indirect – but all the more noteworthy – hints came from quantum field theory, where it was observed that a generalization of symplectic geometry to multisymplectic geometry of degree nn more naturally captures the description of nn-dimensional QFT (notice that quantum mechanics may be understood as (0+1)(0+1)-dimensional QFT). For, observe that the symplectic form on a symplectic Lie n-algebroid is, while always “binary”, nevertheless a representative of de Rham cohomology in degree (n+2)(n+2).

This has an evident generalization to closed forms of degree (n+2)(n+2). If integral, these may be refined to a curvature(n+2)(n+2)-form on a circle n-bundle with connection . Since in the context of smooth ∞-groupoids we can have circle nn-bundles over other smooth ∞\infty-groupoids, this means that we canonically have the notion of prequantum circle (n+1)(n+1)-bundles on a symplectic nn-groupoid.

We observe below that this condition is equivalent to the fact that the flowexp(v):X→X\exp(v) : X \to X of vv preserves the connection on any prequantum line bundle, up to homotopy (up to gauge transformation). In this form the definition has an immediate generalization to symplectic nn-groupoids.

This says that for vv to be Hamiltonian, its contraction with ω\omega must be exact. This is precisely the definition of Hamiltonian vector fields. The corresponding Hamiltonian here is α′−ιvA\alpha'-\iota_v A.